OSGOOD TYPE REGULARITY CRITERION FOR THE 3D NEWTON-BOUSSINESQ EQUATION

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1 Electronic Journal of Differential Equations, Vol. 013 (013), No. 3, pp ISSN: URL: or ftp ejde.math.txstate.edu OSGOOD TYPE REGULARITY CRITERION FOR THE 3D NEWTON-BOUSSINESQ EQUATION ZUJIN ZHANG, SADEK GALA Abstract. In this article, we show an Osgood type regularity criterion for the three-dimensional Newton-Boussinesq equations, which improves the recent results in [4]. 1. Introduction In this article, we consider the three-dimensional Newton-Boussinesq equation ω t + (u )ω ω = (θe 3 ), θ t + (u )θ θ = 0, u = 0, u(0) = u 0, θ(0) = θ 0, (1.1) where ω = u, and u is the velocity field, θ is the scalar temperature, while u 0, θ 0 are the prescribed initial data with u 0 = 0 in distributional sense. System (1.1) arises from the study of Bénard flow [1]. Guo [, 3] investigated the two-dimensional (D) periodic case by using spectral methods and nonlinear Galerkin methods. Meanwhile, the existence and regularity of a global attractor for the D Newton-Boussinesq equations were obtained in [5]. Consequently, it is desirable to consider the regularity criteria for (1.1). Noticing that the convective term (u )u, (u )θ are the same as that in the 3D Boussinesq equations u t + (u )u u + π = θe 3, θ t + (u )θ θ = 0, u = 0, u(0) = u 0, θ(0) = θ 0, we could prove many regularity conditions as that for the Boussinesq equations. For the 3D Boussinesq equations, Ishimura nad Morimoto [6] showed that if u L 1 (0, T ; L ( )), (1.) 000 Mathematics Subject Classification. 35B65, 76B03, 76D03. Key words and phrases. Newton-Boussinesq equations; regularity criterion; Osgood type. c 013 Texas State University - San Marcos. Submitted June 6, 013. Published October 11,

2 Z. ZHANG, S. GALA EJDE-013/3 then the solution is smooth on (0, T ). Fan and Zhou [7] established the regularity of the solution provided that ω = u L 1( 0, T ; Ḃ0, ( ) ), (1.3) where B 0, ( ) is the homogeneous Besov spaces which will be introduced in Section. The interested readers can find more result in [8, 9] and references cited therein. For the 3D Newton-Boussinesq equations (1.1), Guo and Gala [4] obtained some regularity criteria in terms of Morrey spaces and Besov spaces. One of them reads ω = u L 1 (0, T ; Ḃ0, ( )). (1.4) A blow-up criterion for the D Newton-Boussinesq equations was established in [10]. As we know, Osgood type conditions play an important role in solving uniqueness of solutions to the incompressible fluid equations. Motivated by the recent result [1] for the 3D MHD equations u t u + (u )u ( ) + π =, t + (u ) ( )u =, u = = 0, u(0) = u 0, (0) = 0, we would like to improve (1.4). Precisely, we will prove the following theorem. Theorem 1.1. Let (u 0, θ 0 ) H 1 ( ) with u 0 = 0 in distributional sense. Assume that T sup S q u L dτ <, (1.5) q 0 with S q = q l= q l, l being the Fourier localization operator. Then the solution pair (u, θ) to (1.1) with initial data (u 0, θ 0 ) is smooth on [0, T ]. Remark 1.. Since S q u L 1 q l u L C u Ḃ0,, l= q we indeed improve the regularity condition (1.4) established in [4]. Remark 1.3. When θ = 0, (1.1) reduces to the Navier-Stokes equations, thus our result covers the case for the Navier-Stokes equations. The rest of this article is organized as follows. In Section, we recall the definition of Besov spaces, and some interpolation inequalities. Section 3 is devoted to proving Theorem Preliminaries Let S ( ) be the Schwartz class of rapidly decreasing functions. For f S ( ), its Fourier transform F f = ˆf is defined by ˆf(ξ) = f(x)e ix ξ dx.

3 EJDE-013/3 OSGOOD TYPE REGULARITY CRITERION 3 Let us choose a nonnegative radial function ϕ S ( ) such that { 1, if ξ 1, 0 ˆϕ(ξ) 1, ˆϕ(ξ) = 0, if ξ, and let ψ(x) = ϕ(x) 3 ϕ(x/), ϕ j (x) = 3j ϕ( j x), ψ j (x) = 3j ψ( j x), j Z. For j Z, the Littlewood-Paley projection operators S j and j are, respectively, defined by S j f = ϕ j f, j f = ψ j f. Observe that j = S j S j 1. Also, it is easy to check that if f L ( ), then S j f 0, as j ; S j f f, as j +, in the L sense. By telescoping the series, we thus have the following Littlewood- Paley decomposition f = + j= j f, (.1) for all f L ( ), where the summation is the L sense. Note that j f = j+ l=j l j f = j+ l=j ψ l ψ j f, then from Young s inequality, it readily follows that j f L q C 3j(1/p 1/q) j f L p, (.) where 1 p q, and C is an absolute constant independent of f and j. Let < s <, 1 p, q, the homogeneous Besov space Ḃs p,q is defined by the full-dyadic decomposition such as Ḃp,q s = {f Z ( ); f Ḃs < }, p,q where f Ḃs p,q = { js } + j f L p j= l q, and Z ( ) is the dual space of Z ( ) = {f S ( ); D α ˆf(0) = 0, α N 3 }. Also, it is well-known that Ḣ s ( ) = Ḃs,( ), s R. (.3) We refer the reader to [11] for more detailed properties.

4 4 Z. ZHANG, S. GALA EJDE-013/3 3. Proof of Theorem 1.1 This section is devoted to proving Theorem (1.1). Taking the inner products of (1.1) 1, (1.1) with ω, θ in L ( ) respectively, we have 1 d dt ω L + ω L = (θe 3 ) ω dx, R 3 1 d dt θ L + θ L = [(u )θ] θ dx. Adding together yields 1 d dt [ ω L + θ L ] + [ ω L + θ L ] = (θe 3 ) ω dx + [(u )θ] θ dx R 3 θ L ω L [( u )θ] θ dx 1 θ L + 1 ω L [( u )θ] θ dx. (3.1) We are now in a position to estimate I = [( u )θ] θ dx. (3.) Applying the Littlewood-Paley decomposition as in (.1), u = q u + u, (3.3) l= q u + where q is a positive integer to be determined later on. Substituting (3.3) in I, we see that I q l u θ dx + l u θ dx l= q + l u θ (3.4) dx For I 1, we have I 1 + I + I 3. I 1 l u L θ L C ( C 3l/ l u L θ L (by (.)) 3l ) 1/ ( C 3q/ u L θ L (by (.3)) = [C q/ θ L ] 3. l u L ) 1/ θ L (3.5)

5 EJDE-013/3 OSGOOD TYPE REGULARITY CRITERION 5 For I, we have I = S q u θ dx S q u L θ L. (3.6) Finally, for I 3, we have I 3 l u L 3 θ L 3 C 1/ l u L θ L θ L by (.) and Gagliardo-Nireberg inequality ( C l ) 1/ ( l ) 1/ θ L l u L θ [C q/ θ L ][ ω L + θ L] (by (.3)). L (3.7) Gathering (3.5), (3.6) and (3.7) together, and plugging them into (3.8), we deduce I [C q/ θ L ] 3 + S q u L θ L + [C q/ θ L ] [ ω L + θ L ]. (3.8) Substituting (3.8) into (3.1), we find d dt [ ω L + θ L ] + [ ω L + θ L ] θ L + [C q/ θ L ] 3 + S q u L θ L + [C q/ θ L ][ ω L + θ L ]. (3.9) Taking q = [ ln ln+ (C θ L )] + 3, where [t] is the largest integer smaller that t R, and ln + t = ln(e + t), then (3.9) implies that d dt [ ω L + θ L ] + 1 [ ω L + θ L ] θ L + C + S q u L Applying Gronwall inequality three times, we deduce ln + ( θ L ) ln + ln + ( θ L ) θ L. t [ ω L + θ L ] + [ ω L + θ L] dτ ( t C exp exp exp 0 0 S q u L ) dτ. By (1.5), the solutions (u, θ) are uniformly bounded in L (0, T ; H 1 ( )), and thus smooth. This completes the proof of Theorem 1.1. Acknowledgements. Zujin Zhang was partially supported by the (Youth) Natural Science Foundation of Jiangxi Province (013BAB11007, 01BAB01014), the Science Foundation of Jiangxi Provincial Department of Education (GJJ13658, GJJ13659), the National Natural Science Foundation of China ( ).

6 6 Z. ZHANG, S. GALA EJDE-013/3 References [1] S. Chen; Symmetry analysis of convection patterns, Commu. Theor. Phys., 1 (198), [] B. L. Guo; Spectal method for solving two-dimensional Newton-Boussinesq equation, Acta. Math. Appl. Sin., 5 (1989), [3] B. L. Guo; Galerkin methods for solving two-dimensional Newton-Boussinesq equations, Chin. Ann. Math., 16 (1995), [4] Z. G. Guo, S. Gala; Regularity criterion of the Newton Boussinesq equations in, Commu. Pure Appl. Anal., 11 (01), [5] G. Fucci, B. Wang, P. Singh; Asymtotic behavior of the Newton-Boussinesq equations in a two-dimensional channel, Nonlinear Anal., 70 (009), [6] N. Ishihara, H. Morimoto; Remarks on the blow-up criterion for the 3D Boussinesq equations, Math. Models Methods Appl., 9 (1999), [7] J. S. Fan, Y. Zhou; A note on the regularity criterion for the 3D Boussinesq equations with partial viscosity, Appl. Math. Lett., (009), [8] H. Qiu, Y. Du, Z.A. Yao; Blow-up criteria for the 3D Boussinesq equations in the multiplier spaces, Commun. Nonlinear Sci. Numer. Simul., 16 (011), [9] H. Qiu, Y. Du, Z.A. Yao; Serrin-type blow-up criteria for three-dimensional Boussinesq equations, Appl. Anal., 89 (010), [10] H. Qiu, Y. Du, Z.A. Yao; A note on the regularity criterion of the two-dimensional Newton- Boussinesq equations, Nonlinear Anal., Real World Appl., 1 (011), [11] H. Triebel; Interpolation theory, function spaces, differential operators, North Holland, Amsterdam, New-York, Oxford, [1] Q. Zhang; Refined blow-up criterion for the 3D magnetohydrodynamics equations, Appl. Anal., doi: / Zujin Zhang School of Mathematics and Computer Science, Gannan Normal University, Ganzhou , China address: zhangzujin361@163.com Sadek Gala Department of Mathematics, University of Mostaganem, Box 7, Mostaganem, 7007, Algeria address: sadek.gala@gmail.com